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Euclid set a precedent for the codification of mathematics by axiomatizing the set of geometric truths. An obvious question that arises is whether all branches of mathematics are axiomatizable, especially fundamental ones, such as arithmetic. In the late nineteenth century, what became known as Peano arithmetic was proposed as an axiomatization. The essential feature of an axiomatization is that, although one might have an infinite number of axioms, as does Peano arithmetic, one must have a decision procedure for determining whether a given proposition is or is not an axiom. In 1931, Gödel proved the astounding result that, not only is Peano arithmetic incomplete in the sense that it does not entail all arithmetic truths, but any attempted axiomatization of arithmetic is incomplete, and thus the set of arithmetic truths must be undecidable. Subsequently, Alfred Tarski showed the set of arithmetic truths is not even definable. Also, by finding a finitely axiomatizable undecidable subtheory of Peano arithmetic, Alonzo Church was able to show that there is not even an effective procedure for determining whether a given sentence is a logical truth. Finally, in his 1931 paper, Gödel argued a second incompleteness theorem, viz., that any theory strong enough to express its own consistency, as he showed Peano arithmetic to be, cannot prove its own consistency unless it is inconsistent. We will cover these and other results that have had a profound effect on the foundations of mathematics. It remains an open question whether so basic a theory as Peano arithmetic is consistent.